Chapter 1

Use the concept that \(y=c,-\infty

A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water, and water leaks from a circular hole of radius \(\frac{1}{2}\) inch at its bottom, determine a differential equation for the height \(h\) of the water at time \(t\). Ignore friction and contraction of water at the hole.

Use the concept that \(y=c,-\infty

Use the concept that \(y=c,-\infty

In Problems 37 and 38 , verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\) $$ \begin{aligned} &\frac{d x}{d t}=x+3 y \\ &\frac{d y}{d t}=5 x+3 y \\ &x=e^{-2 t}+3 e^{6 t} \\ &y=-e^{-2 t}+5 e^{6 t} \end{aligned} $$

Verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\). $$ \begin{aligned} &\frac{d x}{d t}=x+3 y \\ &\frac{d y}{d t}=5 x+3 y \\ &x=e^{-2 t}+3 e^{6 t} \\ &y=-e^{-2 t}+5 e^{6 t} \end{aligned} $$

In Problems 37 and 38 , verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\) $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}=4 y+e^{t} \\ &\frac{d^{2} y}{d t^{2}}=4 x-e^{t} \\ &x=\cos 2 t+\sin 2 t+\frac{1}{5} e^{t} \\ &y=-\cos 2 t-\sin 2 t-\frac{1}{5} e^{t} \end{aligned} $$

Make up a differential equation that does not possess any real solutions.

Suppose that \(P^{\prime}(t)=0.15 P(t)\) represents a mathematical model for the growth of a certain cell culture, where \(P(t)\) is the size of the culture (measured in millions of cells) at time \(t\) (measured in hours). How fast is the culture growing at the time \(t\) when the size of the culture reaches 2 million cells?

Population Dynamics Suppose that \(P^{\prime}(t)=0.15 P(t)\) represents a mathematical model for the growth of a certain cell culture, where \(P(t)\) is the size of the culture (measured in millions of cells) at time \(t\) (measured in hours). How fast is the culture growing at the time \(t\) when the size of the culture reaches 2 million cells?

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y(0)=0, y(\pi / 6)=-1 $$

Make up a differential equation that you feel confident possesses only the trivial solution \(y=0 .\) Explain your reasoning.

Suppose that $$ A^{\prime}(t)=-0.0004332 A(t) $$ represents a mathematical model for the decay of radium 226 , where \(A(t)\) is the amount of radium (measured in grams) remaining at time \(t\) (measured in years). How much of the radium sample remains at time \(t\) when the sample is decaying at a rate of \(0.002\) grams per year?

Radioactive Decay Suppose that $$ A^{\prime}(t)=-0.0004332 A(t) $$ represents a mathematical model for the decay of radium226 , where \(A(t)\) is the amount of radium (measured in grams) remaining at time \(t\) (measured in years). How much of the radium sample remains at time \(t\) when the sample is decaying at a rate of \(0.002\) grams per year?

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ v(0)=0, v(\pi)=0 $$

What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple \(k\) of itself? Write each answer in the form of a first-order differential equation with a solution.

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y^{\prime}(0)=0, y^{\prime}(\pi / 4)=0 $$

What function (or functions) do you know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution.

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y(0)=1, y^{\prime}(\pi)=5 $$

Given that \(y=\sin x\) is an explicit solution of the first-order differential equation \(d y / d x=\sqrt{1-y^{2}} .\) Find an interval \(I\) of definition. [Hint: \(I\) is not the interval \((-\infty, \infty) .]\)

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y(0)=0, y(\pi)=4 $$

Discuss why it makes intuitive sense to presume that the linear differential equation \(y^{\prime \prime}+2 y^{\prime}+4 y=5 \sin t\) has a solution of the form \(y=A \sin t+B \cos t\), where \(A\) and \(B\) are constants. Then find specific constants \(A\) and \(B\) so that \(y=A \sin t+B \cos t\) is a particular solution of the \(\mathrm{DE}\).

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y^{\prime}(\pi / 3)=1, y^{\prime}(\pi)=0 $$

Find a function \(y=f(x)\) whose graph at each point \((x, y)\) has the slope given by \(8 e^{2 x}+6 x\) and has the \(y\) -intercept \((0,9)\).

Find a function \(y=f(x)\) whose second derivative is \(y^{\prime \prime}=\) \(12 x-2\) at each point \((x, y)\) on its graph and \(y=-x+5\) is tangent to the graph at the point corresponding to \(x=1\).

The given figure represents the graph of an implicit solution \(G(x, y)=0\) of a differential equation \(d y / d x=f(x, y)\). In each case the relation \(G(x, y)=0\) implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution \(\phi\) must be a function and differentiable. Use the solution curve to estimate the interval \(I\) of definition of each solution \(\phi\).

The graphs of the members of the one-parameter family \(x^{3}+\) \(y^{3}=3 c x y\) are called folia of Descartes. Verify that this family is an implicit solution of the first-order differential equation $$ \frac{d y}{d x}=\frac{y\left(y^{3}-2 x^{3}\right)}{x\left(2 y^{3}-x^{3}\right)} $$

Determine a plausible value of \(x_{0}\) for which the graph of the solution of the initial-value problem \(y^{\prime}+2 y=3 x-6, y\left(x_{0}\right)=0\) is tangent to the \(x\) -axis at \(\left(x_{0}, 0\right)\). Explain your reasoning.

Population Growth Beginning in the next section we will see that differential equations can be used to describeor model many different physical systems. In this problem, suppose that a model of the growing population of a small community is given by the initial-value problem $$ \frac{d P}{d t}=0.15 P(t)+20, \quad P(0)=100 $$ where \(P\) is the number of individuals in the community and time \(t\) is measured in years. How fast, that is, at what rate, is the population increasing at \(t=0 ?\) How fast is the population increasing when the population is 500 ?

Find a linear second-order differential equation \(F\left(x, y, y^{\prime}, y^{\prime \prime}\right)=0\) for which \(y=c_{1} x+c_{2} x^{2}\) is a two- parameter family of solutions. Make sure that your equation is free of the arbitrary parameters \(c_{1}\) and \(c_{2}\).

Find a linear second-orderdifferential equation \(F\left(x, y, y^{\prime}, y^{\prime \prime}\right)=0\) for which \(y=c_{1} x+c_{2} x^{2}\) is a two- parameter family of solutions. Make sure that your equation is free of the arbitrary parameters \(c_{1}\) and \(c_{2}\).

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems \(55-58\), recall the geometric significance of the derivatives \(d y / d x\) and \(d^{2} y / d x^{2}\). Consider the differential equation \(d y / d x=e^{-x^{2}}\). (a) Explain why a solution of the DE must be an increasing function on any interval of the \(x\) -axis. (b) What are \(\lim _{x} d y / d x\) and \(\lim d y / d x ?\) What does this \(x \rightarrow \infty\) suggest about a solution curve as \(x \rightarrow \pm \infty\) ? (c) Determine an interval over which solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)-(c).

Consider the differential equation \(d y / d x=e^{-x^{2}}\). (a) Explain why a solution of the DE must be an increasing function on any interval of the \(x\)-axis. (b) What are \(\lim _{x \rightarrow-\infty} d y / d x\) and \(\lim _{x \rightarrow \infty} d y / d x\) ? What does this suggest about a solution curve as \(x \rightarrow \pm \infty\) ? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)-(c).

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems \(55-58\), recall the geometric significance of the derivatives \(d y / d x\) and \(d^{2} y / d x^{2}\). Consider the differential equation \(d y / d x=5-y\). (a) Either by inspection or by the method suggested in Problems \(33-36\), find a constant solution of the \(\mathrm{DE}\). (b) Using only the differential equation, find intervals on the \(y\) -axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on the \(y\) -axis on which \(y=\phi(x)\) is decreasing. (c) Explain why \(y=0\) is the \(y\) -coordinate of a point of inflection of a solution curve. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)-(c).

Consider the differential equation \(d y / d x=5-y\). (a) Either by inspection, or by the method suggested in Problems 33-36, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the \(y\)-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on the \(y\)-axis on which \(y=\phi(x)\) is decreasing.

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems \(55-58\), recall the geometric significance of the derivatives \(d y / d x\) and \(d^{2} y / d x^{2}\). Consider the differential equation \(d y / d x=y(a-b y)\), where \(a\) and \(b\) are positive constants. (a) Either by inspection, or by the method suggested in Problems 33-36, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the \(y\) -axis on which a nonconstant solution \(y=\phi(x)\) is increasing. On which \(y=\phi(x)\) is decreasing. (c) Using only thedifferentialequation, explain why \(y=a / 2 b\) is the \(y\) -coordinate of a point of inflection of the graph of a nonconstant solution \(y=\phi(x)\). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the \(x y\) -plane into three regions. In each region, sketch the graph of a nonconstant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c).

Consider the differential equation \(d y / d x=y(a-b y)\), where \(a\) and \(b\) are positive constants. (a) Either by inspection, or by the method suggested in Problems 33-36, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the \(y\)-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. On which \(y=\phi(x)\) is decreasing. (c) Using only the differential equation, explain why \(y=a / 2 b\) is the \(y\)-coordinate of a point of inflection of the graph of a nonconstant solution \(y=\phi(x)\). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the \(x y\)-plane into three regions. In each region, sketch the graph of a nonconstant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c).

Consider the differential equation \(y^{\prime}=y^{2}+4\). (a) Explain why there exist no constant solutions of the DE. (b) Describe the graph of a solution \(y=\phi(x)\). For example, can a solution curve have any relative extrema? (c) Explain why \(y=0\) is the \(y\)-coordinate of a point of inflection of a solution curve. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)-(c).